Maximizing SystemLifetime in Wireless Sensor Networks

Qunfeng Dong

Department of Computer Sciences

University of Wisconsin

Madison,WI 53706

qunfeng@cs.wisc.edu

ABSTRACT

Maximizing system lifetime in battery-powered wire-

less sensor networks with power aware topology control

protocols and routing protocols has received intensive

research.In the past,this problem has been mostly

studied from the indirect perspective of energy conser-

vation.Although this leads to solutions that help ex-

tend network lifetime,energy conservation is not the

same problem as network lifetime maximization.Some

researchers have formally studied network lifetime max-

imization problems,based on the assumption that en-

ergy is only consumed by packet transmission.How-

ever,it is well known that in many cases energy is sig-

niﬁcantly consumed during overhearing and idle peri-

ods.In this paper,we try to present formal analysis

of a variety of network lifetime maximization problems

in diﬀerent energy consumption models.In particular,

we identify diﬀerent energy consumption models,deﬁne

a variety of fundamental network lifetime maximization

problems in individual energy consumption models,and

formally analyze their complexity.Polynomial time al-

gorithms are presented for tractable problems,and NP-

hardness proofs are presented for intractable problems.

1.INTRODUCTION

Multi-hop,ad hoc,wireless sensor networks (WSNs)

are considered a promising technology to change our

physical environment and hence our life in this envi-

ronment.WSNs are typically deployed using battery-

powered stationary sensor nodes equipped with sensing,

computing and wireless communicating modules.In a

broad range of potential applications,inexpensive sen-

sors can be embedded into buildings or scattered into

spaces to collect,process,store and send out relevant

information for various civilian or military purposes.

When a data sink (e.g.a base station) is out of reach of

a data source sensor node,they can rely on intermediate

sensor nodes to relay data packets.

A salient feature of battery-powered WSN is its ex-

tremely constrained source of energy supplied by bat-

teries coming with sensor nodes,because sensor nodes

are typically small and thus use tiny batteries.In many

scenarios,it seems infeasible to replace or recharge bat-

teries of sensor nodes.For example,NASA plans to

deploy sensor networks in areas of interest on Mars [1].

Meanwhile,in WSNs,wireless communication is con-

sidered much more energy consuming than sensing and

computing [2].All these factors make it essential to de-

velop eﬃcient routing and topology control protocols to

maintain requested network properties (e.g.connectiv-

ity) for as long a network lifetime as possible.

In the literature,there have been two diﬀerent ap-

proaches to maximizing network lifetime.One indirect

approach aims to minimize energy consumption,while

the other approach directly aims to maximize network

lifetime.Although the indirect approach can help ex-

tend network lifetime,it does not address precisely the

problem of maximizing network lifetime.Therefore,

some researchers have aimed to directly maximize net-

work lifetime.

• Chang and Tassiulas [3,4] considered the problem

of maximizing the time to the ﬁrst node failure for

a unicast session,where each data source generates

data for delivery at a ﬁxed rate.

• In [5,6],optimal solutions are presented for maxi-

mizing the time to the ﬁrst node failure for a static

broadcast tree.

• In the more general multicast paradigm,Das et

al.[7] presented an optimal solution for maximiz-

ing the time to the ﬁrst node failure for a static

multicast tree.Flor´een et al.[8] investigated the

problem of maximizing the lifetime of a multicast

session over a network of energy constrained nodes,

where the multicast tree can be dynamically ad-

justed to utilize any node with available energy.

While these eﬀorts are based on the energy consumption

model where energy is consumed only when transmit-

ting packets,it is well known that wireless transceivers

consume a signiﬁcant amount of energy during over-

hearing and idle periods as well [9,10,11].

The contribution of this paper is the formal analysis

of a number of network lifetime maximization problems,

under diﬀerent energy consumption models.In par-

ticular,we identify representative energy consumption

models,deﬁne a variety of fundamental network life-

time maximization problems under these models,and

formally analyze their complexity.Polynomial time al-

gorithms are presented for tractable problems,and NP-

hardness proofs are presented for intractable problems.

Despite signiﬁcant research in this area,we do not know

of any optimal solutions to these fundamental problems

identiﬁed in this paper,and the complexity of these

problems remain unknown.To the best of our knowl-

edge,this paper is the ﬁrst to present such a formal

analysis.

The rest of the paper is organized as follows.In Sec-

tion 2,we identify representative energy consumption

models and deﬁne network lifetime in individual en-

ergy consumption models.In Section 3,various net-

work lifetime maximization problems are deﬁned under

individual energy consumption models.The complex-

ity of these problems is formally analyzed.Finally,we

conclude the paper in Section 4.

2.MODELS AND DEFINITIONS

In most of the past research eﬀorts aiming to extend

network lifetime,energy consumption is completely at-

tributed to packet transmission.Wireless transceivers

are assumed to consume power only when transmitting

packets,and energy is thus consumed on a per packet

basis.This model is simple and neat.In this paper,

we also include this energy consumption model in our

analysis.For simplicity,we refer to this model as the

packet based model.

Despite the prevalence of the packet based model,it

has been well known that energy is also signiﬁcantly

consumed during overhearing and idle periods [9,10,

11].In particular,wireless transceivers are powered

to receive every incoming packet and decode to decide

if the packet should be accepted,forwarded,or dis-

carded.Although many packets turn out to be sim-

ply discarded,their reception has already consumed

a signiﬁcant amount of energy.In addition,wireless

transceivers also consume energy during idle periods,

because they have to be powered to detect if there are

packets being transmitted at all.Researchers [9,11]

have shown that in some cases,energy consumption

during overhearing and idle periods can be comparable

to energy consumption due to transmitting/receiving

packets.In many applications,WSNs are presumed

to be densely deployed,and this has two implications.

On one hand,pair-wise distance between sensor nodes

is small,and thus packet transmission between sensor

nodes consumes less energy.On the other hand,each

sensor node covers more sensor nodes in its transmis-

sion range,and thus more energy will be consumed due

to overhearing.

In the extreme case where wireless transceivers stay

idle and no communication happens at all,energy is

completely consumed in the idle state,on a per time

unit basis.We hereby refer to this energy consumption

model as the time based model.In a broad range of ap-

plications where sensor nodes only need sporadic (and

possibly asynchronous) communication,power consump-

tion is dominated by idle time and transceivers consume

almost the same amount of energy.For example,sen-

sor nodes may be conﬁgured to send back environment

information once per hour.The time based model ﬁts

well into such scenarios.In such scenarios,to eﬀec-

tively conserve energy and extend network lifetime,it

is no longer adequate to simply optimize transmission

power as has been done by most researchers.Instead,

we need to turn oﬀ as many transceivers as much as

possible.When a sensor node’s transceiver is turned

oﬀ,it is considered sleeping.In the sleeping state,en-

ergy consumption during overhearing and idle periods

is avoided.Communication is handled by a backbone

composed of nodes that do not sleep,connecting every

pair of nodes in the network.Asleeping node may occa-

sionally wake up to send out packets over the backbone.

That part of the energy consumption can be addressed

by the packet based model.

In cases where communication is relatively frequent,

energy consumption can be divided into two parts.On

one hand,(homogeneous) sensor nodes consume as much

power as each other on a per time unit basis,due to

overhearing and staying idle.On the other hand,they

may consume signiﬁcantly diﬀerent amounts of energy

on a per packet basis,due to packet transmission/reception.

We refer to this case as the mixed model.

Various deﬁnitions of network lifetime have been pro-

posed for diﬀerent scenarios.In [12],Blough and Santi

present a discussion on deﬁning network lifetime,and

outline the principle that network lifetime should refer

to the capability of the network to serve its design pur-

pose.In this paper,we deﬁne network lifetime for a

number of network lifetime maximization problems ac-

cording to this general principle.For problems in the

packet based model,we deﬁne network lifetime as the

number of packets (to be perfectly accurate,the number

of bits) that can be delivered by the network.This def-

inition applies to all routing paradigms including uni-

cast,multicast and broadcast.

In the time based model,the design purpose is to

maintain an always active communication backbone con-

necting every pair of nodes in the network.Accordingly,

we deﬁne network lifetime to be the time until no such

backbone can be formed.This deﬁnition is also moti-

vated by the following features of WSNs.

• On one hand,sensor nodes are presumed to be

densely deployed and sensor networks are thus highly

redundant.Even if some sensor nodes fail due

to battery depletion,the whole sensor network is

most likely still in good order to serve its purpose.

• On the other hand,wireless communication is con-

sidered the primary cause of energy consumption

in WSNs,especially in many applications where

sensor nodes only need to conduct modest data

collecting and processing.Even if this assump-

tion is not true in some cases,we may reserve a

certain amount of energy for sensing,processing

and sending data,and reserve the rest of avail-

able energy for staying active in the backbone and

relaying packets.Thus,even if some sensor node

has run out of its energy for relaying packets,it can

still collect,process and send out data as usual.As

long as there exists such a backbone,the function-

ality of the whole sensor network remains intact to

serve its design purpose.

3.ANALYSIS

In this section,we formally analyze a variety of net-

work lifetime maximization problems in the time based

model as well as the intensively researched packet based

model.Deﬁnitions and complexity analysis of problems

speciﬁc to individual models are presented.Note that

the time based model and the packet based model are

both special cases of the mixed model,thus their hard-

ness results trivially apply to the mixed model.

In our network model,stationary sensor nodes are as-

sumed to be equipped with an omnidirectional antenna.

A wireless sensor network is denoted by a weighted di-

rected graph G = (V,A),where V is the set of sensor

nodes and A is the set of directed links.Each node is

labeled with a unique ID i ∈ [1..|V |] and has a maxi-

mum transmission power of P

i

max

.Let P

ij

denote the

minimum transmission power required to maintain a

reasonably good quality link from node i to node j.G

contains link (i,j) (i.e.,the link from node i to node j)

if and only if P

ij

≤ P

i

max

.Initially,each sensor node

i ∈ V has an energy of p

i

.Time is divided into discrete

time slots,denoted by t ≥ 1,t ∈ Z

+

.

3.1 The time based model

In this section,we investigate the problem of max-

imizing network lifetime in the time based model and

prove it to be NP-hard.When proving the NP-hardness

of intractable problems identiﬁed in this section,we ac-

tually prove stronger results that the problems remain

NP-hard even if they are restricted to the special case

where all sensor nodes have the same maximum trans-

mission power P

max

and P

ij

= P

ji

for each node pair

(i,j).In this case,a stationary wireless sensor net-

work can be modelled as a weighted undirected graph

G = (V,E),where E is the set of undirected edges and

G contains edge (i,j) if and only if P

ij

≤ P

max

.

To maintain network connectivity,what we need is

a backbone,which is represented as a connected domi-

nating set (CDS) [13].In an undirected graph G(V,E),

a dominating set is deﬁned as a subset S ⊆ V of nodes

such that each node i ∈ V is either in S or adjacent

to some node v ∈ S.A connected dominating set S

is a dominating set such that the subgraph G

′

= (S ⊆

V,E

′

⊆ E) induced by S is connected.Here,we shall

prove an even stronger result that the problemof maxi-

mizing network lifetime while preserving connectivity in

undirected graphs remains NP-hard even if we restrict

it to the special case where during each time step,each

node i ∈ V consumes p

i

energy.In this case,each

node has a battery life of one (time slot) and can be

used in exactly one CDS.The problem of maximizing

network lifetime thus becomes the connected domatic

number (CDN) problem,which is deﬁned as follows.

Connected Domatic Number (CDN)

INSTANCE Graph G = (V,E).Positive

integer K.

QUESTION Does G contain at least K

disjoint CDSs?

Theorem 1.Connected domatic number is NP-hard.

Proof.We prove the NP-hardness of CDNby reduc-

ing from the 3-dimensional matching (3DM) problem,

which is known to be NP-hard [14] and formally deﬁned

as follows.

3-Dimensional Matching (3DM)

INSTANCE Set M = {m

1

,m

2

,...,m

m

} ⊆

W × X × Y,where W = {w

1

,w

2

,...,w

q

},

X = {x

1

,x

2

,...,x

q

},and Y = {y

1

,y

2

,...,y

q

}

are disjoint sets having the same number q of

elements and |M| = m.

QUESTION Does M contain a matching,

i.e.,a subset M

′

= {m

′

1

,m

′

2

,...,m

′

q

} ⊆ M

such that |M

′

| = q and no two elements of

M

′

agree in any coordinate?

Given an instance of 3DM,we construct a graph G =

(V,E) as shown in Fig.2,where nodes are distributed

into four layers and edges exist only between nodes in

the same layer or adjacent layers.The graph in Fig.2

is constructed from the following instance of 3DM.

W = {w

1

,w

2

},X = {x

1

,x

2

},Y = {y

1

,y

2

}

and M = {m

1

,m

2

,m

3

,m

4

},where m

1

=

(w

1

,x

2

,y

1

),m

2

= (w

1

,x

1

,y

1

),m

3

= (w

2

,x

2

,y

2

),

and m

4

= (w

2

,x

1

,y

2

).

In the top layer,there are 3 disjoint groups of set

nodes,W = {W

1

,W

2

,...,W

m−q

},X = {X

1

,X

2

,...,X

m−q

},

and Y = {Y

1

,Y

2

,...,Y

m−q

}.In the second layer,there

are 3 corresponding disjoint groups of element nodes,

W = {w

1

,w

2

,...,w

q

},X = {x

1

,x

2

,...,x

q

},and Y =

{y

1

,y

2

,...,y

q

}.W,X,and Y represent W,X,and Y in

the 3DM instance,respectively.W ∪ W forms a clique

of size m,and so do X ∪ X and Y ∪ Y.Besides the

element nodes,the second layer also contains a group

B = {b

1

,b

2

,...,b

m−q

} of bridge nodes.Each bridge

node is adjacent to every set node in the top layer.In

the third layer,there is a group M= {m

1

,m

2

,...,m

m

}

of triplet nodes representing the elements in M.Each

bridge node in the second layer is adjacent to every

triplet node as well.Each triplet node is also adjacent

to the 3 element nodes that occur in the element in M

that it represents.In the bottom layer,there is a group

M

′

= {m

′

1

,m

′

2

,...,m

′

q

} of matching nodes representing

a potential 3-dimensional matching M

′

.Each matching

node is adjacent to every triplet node in the third layer.

The transformation is clearly polynomial,and we prove

m

′

1

m

′

2

m

1

m

2

m

3

m

4

w

1

w

2

x

1

x

2

y

1

y

2b

1

b

2

W

1

W

2

X

1

X

2

Y

1

Y

2

Figure 1:Reduction from 3DM to CDN.

that M contains a 3-dimensional matching of size q if

and only if G contains m disjoint CDSs.

We start with the “only if” direction.If M contains

a matching of size q,each triplet node in the matching,

its associated element nodes,and a matching node form

a CDS of G.Each of the other m−q CDSs is comprised

of one bridge node,one set node from each of W,X,Y,

and one of the remaining triplet nodes.

We proceed to prove the “if” direction.If G contains

m disjoint CDSs,each CDS must contain exactly one

triplet node because matching nodes are only adjacent

to triplet nodes.

Recall that each one of W ∪ W,X ∪ X,and Y ∪ Y

forms a clique comprised of q element nodes and m−q

set nodes.Since each CDS only contains one triplet

node,it can dominate at most one element node in

each clique via its triplet node.Therefore,in non-trivial

cases where q ≥ 2,each CDS also has to contain at least

one node from each clique as well.On the other hand,

each CDS can have at most one node from each clique

since each clique only has m nodes to be shared by m

CDSs.Clearly,each CDS also contains exactly one node

from each clique.

If a CDS contains a set node,the set node can only be

connected to its triplet node via some bridge node,since

we have proven above that a CDS can not have another

node from the same clique to connect the set node to

its triplet node.Given m−q bridge nodes,it is clear

that at most m−q CDSs can contain a set node.On

the other hand,each CDS can contain at most one set

node fromeach clique,which means at least m−q CDSs

have to contain some set node.Therefore,it must be the

case that there are exactly m−q CDSs each containing

one set node from each clique,while each of the other

q CDSs contains one element node from each clique.

Note that in each of these q CDSs,each element node

has to be directly connected to the triplet node since

there can not be another node from the same clique.

Thus,these q CDSs form a 3-dimensional matching of

size q we need.

3.2 The packet based model

In this section,we analyze network lifetime maxi-

mization problems in the intensively researched packet

based model.In particular,we analyze the complexity

of a number of network lifetime maximization problems

in diﬀerent routing paradigms,i.e.,unicast,multicast

and broadcast.NP-hardness proofs are presented for

intractable problems,and polynomial time algorithms

are given for tractable problems.

We start with the problemof maximizing the lifetime

of a broadcast session over energy constrained WSNs,

which is formally deﬁned as follows.

Broadcast lifetime

INSTANCE Directed graph G = (V,A).

Speciﬁed source s.Positive integer K.

QUESTION Does G have enough power

to broadcast K packets from s to all other

nodes?

The problem of minimum energy broadcast has been

well researched in the literature and proved to be NP-

hard [15,16].However,the complexity of broadcast

lifetime remains open.Flor´een et al.[8] investigated the

problemof maximizing the lifetime of a multicast session

over a network of energy-constrained nodes,where the

network contains some critical nodes that have to be

included in every steiner tree.Therefore,[8] did not

address our problem.

Theorem 2.Broadcast lifetime is NP-hard.

Proof.The NP-hardness of broadcast lifetime can

be proved by slightly adapting the proof of Theorem 1.

In particular,we shall also prove by reducing from the

3-dimensional matching (3DM) problem.

Given an instance of 3DM,we construct a graph G =

(V,E) as shown in Figure 2,where nodes are distributed

into four layers and edges exist only between nodes in

the same layer or adjacent layers.Nodes in each layer

is the same as deﬁned in the proof of Theorem 1.The

only diﬀerence is that,in the bottomlayer,there is only

the source node s,which has an energy of m and is ad-

jacent to all the triplet nodes.Each node other than

s has one unit energy.It is clear that the transfor-

mation is polynomial,and we prove that M contains a

3-dimensonal matching of size q if and only if mpackets

can be broadcast.

We start with the “only if” direction.If M contains a

matching of size q,each triplet node in the matching,its

associated element nodes plus the source node s form a

broadcast tree.Each of the other m−q broadcast trees

is composed of one bridge vertex,one set node for each

of W,X,Y,one triplet node,plus the source node s.All

these broadcast trees are node-disjoint (except s) and s

s

m

1

m

2

m

3

m

4

w

1

w

2

x

1

x

2

y

1

y

2b

1

b

2

W

1

W

2

X

1

X

2

Y

1

Y

2

Figure 2:The graph is constructed from the

following 3DM instance.W = {w

1

,w

2

},X =

{x

1

,x

2

},and Y = {y

1

,y

2

}.M = {m

1

,m

2

,m

3

,m

4

},

where m

1

= (w

1

,x

2

,y

1

),m

2

= (w

1

,x

1

,y

1

),m

3

=

(w

2

,x

2

,y

2

),and m

4

= (w

2

,x

1

,y

2

).

has enough power.Thus,one packet can be broadcast

over each of these m broadcast trees,respectively.

We then prove the “if” direction.If m packets can

be broadcast,it is clear that there have to be m node-

disjoint broadcast trees each containing exactly one triplet

node,since s relies on the triplet nodes to forward its

packets.Consequently,in non-trivial cases where q ≥ 2,

each broadcast tree has to contain at least one node

from each clique to broadcast a packet to the nodes in

the cliques.Since each clique has m nodes,each broad-

cast tree has exactly one node from each clique.If a

broadcast tree contains a set node,the set node can

only be connected to the triplet node of that broadcast

tree via a bridge node,since there can not be another

node from the same clique in the broadcast tree.Given

m − q bridge nodes and 3(m − q) set nodes,it must

be the case that there are m− q broadcast tree each

containing one bridge node and one set node from each

clique.Therefore,each of the other q broadcast trees

contains one element node from each clique and the el-

ement nodes have to be adjacent to the triplet node in

the broadcast tree.These q triplet nodes thus form a

3-dimensional matching of size q.

Similarly,in the problem of multicast lifetime,we

want to maximize the number of packets that can be

multicast from a speciﬁed source s to a speciﬁed group

T of terminals.Since broadcast is just a special case

of multicast,the NP-hardness of multicast lifetime di-

rectly follows.

We then proceed to investigate the problem of maxi-

mizing lifetime of unicast sessions.Similarly,we also de-

ﬁne network lifetime as the maximumnumber of packets

that can be delivered by the network.Because even if

some nodes fail due to battery depletion,the network

may still be able to deliver packets for a unicast session.

There are four diﬀerent cases in unicast:one-to-one

unicast,one-to-many unicast,many-to-one unicast and

many-to-many unicast,of which many-to-many unicast

is the most general case.Meanwhile,there are two dif-

ferent ﬂow models in unicast,i.e.,the multiple commod-

ity model and the single commodity model.In the multi-

ple commodity model,packets to be delivered between

each source-sink pair are considered a separate com-

modity.In the more relaxed single commodity model

that has been previously studied by Chang and Tassiu-

las [3],all packets are considered the same commodity

and each sink is satisﬁed if and only if it receives the

number of packets it requests,no matter which source

sends the packets.The most general case of many-to-

many unicast lifetime is formally deﬁned in each model,

respectively.The deﬁnitions of the other three cases

can be easily induced as special cases of many-to-many

unicast lifetime.

Many-to-many unicast lifetime (mul-

tiple commodity model)

INSTANCE Directed graph G = (V,A).

Speciﬁed set of sources S = {s

1

,s

2

,...,s

m

} ⊆

V and speciﬁed set of sinks D = {t

1

,t

2

,...,t

n

} ⊆

V.Each source s

i

has N

ij

packets to be de-

livered to sink t

j

.Positive integer K.

QUESTION Does Ghave enough power to

deliver K packets?

Many-to-many unicast lifetime (single

commodity model)

INSTANCE Directed graph G = (V,A).

Speciﬁed set of sources S = {s

1

,s

2

,...,s

m

} ⊆

V and speciﬁed set of sinks D = {t

1

,t

2

,...,t

n

} ⊆

V.Each source s

i

has N

s

i

packets to be deliv-

ered and each sink t

j

requests for N

t

j

packets.

Positive integer K.

QUESTION Does Ghave enough power to

deliver K packets?

It is clear that the deﬁnition of many-to-one unicast

lifetime,one-to-many unicast lifetime,and one-to-one

unicast lifetime remain the same in the multiple com-

modity model and the single commodity model.

Lemma 1.One-to-one unicast lifetime is NP-hard.

Proof.It suﬃces to prove the NP-hardness of the

special case where every packet is to be sent from a

source node s to a sink node t.Again,we reduce from

3DM and we illustrate the reduction in Figure 3 with

the same 3DMinstance as used in Figure 2.Nodes are

still distributed into four layers.In the top layer,there

is the sink node t.In the second layer,there are 3 dis-

joint groups of element nodes,W = {w

1

,w

2

,...,w

q

},

s

m

1

m

2

m

3

m

4

w

1

w

2

x

1

x

2

y

1

y

2

t

Figure 3:Reduction from 3DM to one-to-one unicast.

X = {x

1

,x

2

,...,x

q

},and Y = {y

1

,y

2

,...,y

q

}.W,X,

and Y represent W,X,and Y,respectively.Each el-

ement node has an energy of 1 and is adjacent to t.

In the third layer,there is a group of m triplet nodes

M = {m

1

,m

2

,...,m

m

} representing the elements in

M.Each triplet node has an energy of 3 and is adja-

cent to t as well as the three element nodes occurring

in the element in M that it represents.In the bot-

tom layer,there is the source node s with an energy of

m+2q.Each triplet node is also adjacent to s.Edges

between triplet nodes and t have a weight of 3,while

the other edges have a weight of 1.The transformation

is clearly polynomial,and we prove that M contains a

3-dimensional matching of size q if and only if m+2q

packets can be gathered from s to t.

If M contains a 3-dimensional matching of size q,3q

packets can be delivered through the q triplet nodes

in the matching and the 3q element nodes.The other

m−q packets can be delivered through the other m−q

triplet nodes.

If m+2q packets can be delivered from s to t,it is

clear that the only way to achieve that is the same as

described above,since every packet has to be forwarded

by some triplet node.Thus,the q triplet nodes adjacent

to the 3q element nodes form a 3-dimensional matching

of size q.

Since one-to-one unicast lifetime is a special case of

the other three unicast lifetime problems,the following

theorem directly follows.

Theorem 3.In both multiple commodty model and

single commodity model,many-to-many unicast lifetime,

many-to-one unicast lifetime,one-to-many unicast life-

time and one-to-one unicast lifetime are all NP-hard.

Although the unicast lifetime problems are proven to

be NP-hard,it turns out that in cases where each node

i has a ﬁxed transmission power of P

max

(i) (e.g.tiny

sensor nodes may not be able to adjust their transmis-

sion power),we may be able to solve themin polynomial

time.

Theorem 4.If each node has a ﬁxed transmission

power,one-to-one unicast lifetime can be solved in poly-

nomial time.

Proof.Given an instance of one-to-one unicast life-

time,for each node i ∈ V,deﬁne its capacity to be

c

i

= p

i

/P

max

(i),where p

i

is its initial energy.An al-

gorithm for the node-capacitated network ﬂow problem

[17] can be applied to compute the maximum number

of packets that can be delivered from s to t.

Theorem 5.If each node has a ﬁxed transmission

power,many-to-one unicast lifetime can be solved in

polynomial time.

Proof.Given that one-to-one unicast lifetime is poly-

nomially solvable,it suﬃces to reduce many-to-one uni-

cast lifetime to one-to-one unicast lifetime.First of all,

we point out that for many-to-one unicast lifetime,we

can safely assume without loss of generality that t/∈ S.

Given an instance of many-to-one unicast lifetime,we

transform it into an instance of one-to-one unicast life-

time as follows.For each source node s

i

,generate a

mirror node s

′

i

with an energy of n

i

,where n

i

is the

number of packets to be delivered from s

i

to the sink t.

Then,add a directed link of weight 1 from s

′

i

to s

i

.Fi-

nally,add a super source s and a directed link of weight

0 froms to each mirror node.All packets are now to be

delivered froms to t.It is clear that the transformation

is polynomial.

Assume that K packets can be delivered to t in the

given instance of many-to-one unicast lifetime,where

each source node s

i

has n

′

i

≤ n

i

packets delivered to t.

In the constructed instance of one-to-one unicast life-

time,s can safely dispatch the n

′

i

packets to s

i

via s

′

i

,

and all the K packets can be delivered to t along the

same paths as they travel along in the given instance of

many-to-one unicast lifetime.

On the other hand,if K packets can be delivered

from s to t in the constructed one-to-one unicast life-

time instance,each packet has to travel through some

source node s

i

.The available energy at mirror nodes

guarantees that for each 1 ≤ i ≤ m,at most n

i

pack-

ets ﬁrst reaches s

i

among the source nodes.Thus,in

the given instance of many-to-one unicast lifetime,K

packets can travel from the sources to t along the same

paths as they travel along in the constructed instance

of one-to-one unicast lifetime.

Theorem 6.If each node has a ﬁxed transmission

power,one-to-many unicast lifetime can be solved in

polynomial time.

Proof.We similarly prove by reducing to one-to-

one unicast lifetime and point out that for one-to-many

unicast lifetime,we can also safely assume without loss

of generality that s/∈ D.

Given an instance of one-to-many unicast,we trans-

formit into an instance of one-to-one unicast lifetime as

follows.For each sink node t

i

,generate a mirror node t

′

i

with an energy of n

i

,where n

i

is the number of packets

to be delivered from the source node s to t

i

.And add

a directed link of weight 0 from t

i

to t

′

i

.Then,add a

super sink t and a directed link of weight 1 from each

mirror node to t.All packets are now destined to t.It

is clear that the transformation is polynomial.

Assume that K packets can be delivered in the given

instance of one-to-many unicast lifetime,where each

sink node t

i

receives n

′

i

≤ n

i

packets.Then in the con-

structed instance of one-to-one unicast lifetime,each

sink node t

i

can simply forward the n

′

i

packets to t via

t

′

i

,and all the K packets are thus delivered to t.

On the other hand,if K packets can be delivered from

s to t in the constructed instance of one-to-one unicast

lifetime,each packet has to travel through some sink

node t

i

.The available energy at mirror nodes guaran-

tees that for each 1 ≤ i ≤ n,at most n

i

packets travel

to t via t

′

i

.Thus,in the given instance of one-to-many

unicast lifetime,K packets can be delivered along the

same paths as they travel along in the constructed in-

stance of one-to-one unicast lifetime.

Theorem 7.In the single commodity model,if each

node has a ﬁxed transmission power,many-to-many uni-

cast lifetime is polynomially solvable.

Proof.In the single commodity model,packets can

be delivered fromany source to any sink.Thus,we only

need to consider non-trivial cases where S∩D = φ.Re-

call that many-to-one unicast lifetime remains the same

in the multiple commodity model and the single com-

modity model.Given Theorem 5,it suﬃces to reduce

many-to-many unicast lifetime to many-to-one unicast

lifetime.

Given an instance of many-to-many unicast lifetime,

generate a mirror node t

′

i

with an energy of n

i

for each

sink t

i

,where n

i

is the number of packets requested by

t

i

.Add a directed link (t

i

,t

′

i

) of weight 0.Then,add a

super sink t and a directed link of weight 1 from each

mirror node to t.All packets are now destined to t.The

transformation is clearly polynomial.

Assume that K packets can be delivered in the given

instance of many-to-many unicast lifetime,where each

sink node t

i

receives n

′

i

≤ n

i

packets.Then in the

constructed instance of many-to-one unicast lifetime,

each sink node t

i

can simply forward the n

′

i

packets to

t via t

′

i

,and all the K packets are thus delivered to t.

On the other hand,if K packets can be delivered to t

in the constructed instance of many-to-one unicast life-

time,each packet has to travel through some sink node

t

i

.The available energy at mirror nodes guarantees that

for each 1 ≤ i ≤ n,at most n

i

packets travel to t via

t

′

i

.Thus,in the given instance of many-to-many uni-

cast lifetime,K packets can be delivered along the same

paths as they travel along in the constructed instance

of many-to-one unicast lifetime.

Theorem 8.In the multiple commodity model,even

if each node has a ﬁxed transmission power,many-to-

many unicast lifetime remains NP-hard.

Proof.We prove by reducing fromthe NP-hard dis-

joint connecting paths problem [14],which is deﬁned as

follows.

Disjoint connecting paths

INSTANCE Graph G = (V,E),where V

is the set of nodes and E is the set of edges.

Disjoint set of sources S = {s

1

,s

2

,...,s

m

} ⊆

V and set of sinks D = {t

1

,t

2

,...,t

m

} ⊆ V.

QUESTION Does G contain m node dis-

joint paths,each connecting one pair of source

and sink (s

i

,t

i

) for all 1 ≤ i ≤ m?

Given an instance of disjoint connecting paths,assign

each edge a weight of 1.Assign each sink node an energy

of 0 and each non-sink node an energy of 1.Let each

s

i

have one packet to be delivered to the correspond-

ing sink t

i

.The transformation is clearly polynomial,

and we show that m packets can be delivered from the

sources to the sinks if and only if there are m node dis-

joint paths each connecting one pair of source and sink

(s

i

,t

i

) for all 1 ≤ i ≤ m.

If G contains m node disjoint paths each connecting

one pair of source and sink (s

i

,t

i

) for all 1 ≤ i ≤ m,each

s

i

can deliver its packet to t

i

along the path connecting

them.And all of the m packets can thus be delivered.

Assume that all of the m packets can be delivered.

Since each edge has a weight of 1 and non-sink nodes

have an energy of 1,each non-sink node is on the deliv-

ery path of at most one packet.Sink nodes do not have

energy and there is only one packet destined to each

sink,thus each sink node is on the delivery path of at

most one packet as well.Therefore,the delivery paths

of the m packets are node disjoint,each connecting one

pair of source and sink (s

i

,t

i

) for all 1 ≤ i ≤ m.

4.CONCLUSIONS

We have presented formal analysis of a variety of net-

work lifetime maximization problems in diﬀerent en-

ergy consumption models.An analysis of energy con-

sumption in wireless sensor networks leads to two en-

ergy consumption models for formal analysis,i.e.,the

time based model and the intensively researched packet

based model.Various network lifetime maximization

problems are identiﬁed in individual models.The com-

plexity of these problems are formally analyzed.

Most of the past research eﬀorts aiming to extend

network lifetime are based on the packet based model,

while it is well known that in many applications energy

consumption in the time based model is comparable to

that in the packet based model.On the other hand,

there are two diﬀerent approaches to network lifetime

maximization,and most of the past research eﬀorts fol-

lowed the indirect approach of energy conservation.Al-

though helpful to extend network lifetime,energy con-

servation is not precisely the same problem as network

lifetime maximization.

In this paper,we directly investigate the problem of

network lifetime maximization in individual energy con-

sumption models as well as routing paradigms.In the

time based model,we study the problem of maximiz-

ing network lifetime while preserving connectivity and

prove that it is NP-hard.In the packet based model,we

formally deﬁne the following problems:broadcast life-

time,multicast lifetime,many-to-many unicast lifetime,

many-to-one unicast lifetime,one-to-many unicast life-

time and one-to-one unicast lifetime.Broadcast lifetime

and multicast lifetime are NP-hard,even if each node

has a ﬁxed transmission power.We show that the uni-

cast lifetime problems are NP-hard in both the multi-

ple commodity model and the single commodity model.

However,we show that in cases where each node has

a ﬁxed transmission power,many-to-one unicast life-

time,one-to-many unicast lifetime,and one-to-one uni-

cast lifetime are polynomially solvable.Many-to-many

unicast lifetime is also polynomially solvable in the sin-

gle commodity model,but remains NP-hard in the mul-

tiple commodity model.

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